Series in Real Analysis, 1. Singapore: World Scientific Publishing Co.. xii, 206 p. £ 53.00 (1988).

The purpose of this book is to give a detailed study of the theory of integration together with some applications, based on the definition due to {\it J. Kurzweil} [Czech. Math. J. 7(82), 418-446 (1957;

Zbl 0090.300)] and to the author [J. Lond. Math. Soc. 30, 273-286 (1955;

Zbl 0066.092); Proc. Lond. Math. Soc., III. Ser. 11, 402-418 (1961;

Zbl 0099.274)].
In spite of its generality, the definition is surprisingly simple and elementary. Let $E=\prod\sp{n}\sb{1}[a\sb i,b\sb i)$ be a brick in $R\sp n$, and, for each brick $I\subset E$ and each vertex x of I, let h(I,x) be a real (or complex) number. The integral $\int\sb{E}dh$ is defined to be a real (or complex) number H such that, for a given $\epsilon >0$, there is a strictly positive function $\delta$, defined on the closure of E, satisfying $\vert s-H\vert <\epsilon$ whenever $s=\sum\sp{n}\sb{1}h(I\sb k,x\sb k),$ E is the union of the mutually disjoint bricks $I\sb 1,...,I\sb m,$ $x\sb k$ is one of the vertices of $I\sb k$, and $diam I\sb k<\delta (x\sb k).$ It is shown that this integral embrasses, among others, those of Riemann-Stieltjes, Burkill, Lebesgue, Denjoy-Perron, and that all essential properties of the Lebesgue integral can be, under suitable conditions and a suitable form, generalized to it. So integration of sequences of functions, derivation with respect to a parameter, differentiation of the integral function $H(E)=\int\sb{E}f d\mu,$ Fubini and Tonelli-type theorems for integrals in $R\sp{m+n}$ are discussed. Applications in the theory of ordinary differential equations, probability theory and statistics are added. The book ends with a careful survey on relations with other integrals and with detailed historical remarks.